It is fairly easy to show that is strictly decreasing for , and that the limit as is . Which implies that has no roots for and as it is greater than zero as for arbitarily small that proves your result.

RonL

Yeah, that is a well known method of showing that an inequality is true, but can you use a similar method to find that other function.

For example this method shows the inequality, but you cannot not use it to go from

I am sorry, I will state explicitly what I want. Lets say that I was just given and I needed to find some function that is always smaller than it but will also diverge by the integral test. In other words your method shows how to show that the inequality is true. But in only works given the fact taht we know the inequality, say we didnt know the function that was smaller than and diverged by the integral test. How would one go about taking and manipulating it to find such a function.

If that was convoluted just ignore it, there is no need to prove this, its just out of curiosity